Using fluctuating hydrodynamics we describe the slow buildup of long range spatial correlations in a freely evolving fluid of inelastic hard spheres. In the incompressible limit, the behavior of spatial velocity correlations (including r 2d behavior) is governed by vorticity fluctuations only and agrees well with two-dimensional simulations up to 50 to 100 collisions per particle. The incompressibility assumption breaks down beyond a distance that diverges in the elastic limit. [S0031-9007(97) In the characterization of granular matter as an unusual solid, fluid, or gas by Jaeger et al. [1], this Letter addresses the granular gas regime, controlled by inelasticity, clustering [2], and collapse [3]. Clustering is a long wavelength, low frequency (hydrodynamic) phenomenon and inelastic collapse a short wavelength, high frequency (kinetic) phenomenon. In the granular gas regime, also called rapid granular flows, the dynamics is dominated by inelastic collisions. Here the methods of nonequilibrium statistical mechanics, molecular dynamics, kinetic theory, and hydrodynamics are most suitable for describing the observed average macroscopic behavior [2][3][4][5][6][7] and the fluctuations around it.The lack of energy conservation makes the granular gas, whether driven or freely evolving, behave very differently from molecular fluids. The essential physical processes and detailed dynamics are described in [2,3] and references therein: the similarities and differences with molecular fluids; lack of separation of microscales and macroscales, not only because the grains themselves are macroscopic, but also because of the existence of intermediate intrinsic scales which are controlled by the inelasticity and are only well separated when the system is nearly elastic. A simple model which incorporates the inelasticity of the granular collisions consists of inelastic hard spheres (IHS), taken here of unit mass and diameter, with momentum conserving dynamics. The energy loss in a collision is proportional to the inelasticity parameter e 1 2 a 2 where a is the coefficient of normal restitution.For an understanding of what follows, we recall two important properties of the undriven granular gas: (i) the existence of a homogeneous cooling state (HCS) and (ii) its instability against spatial fluctuations. The hydrodynamic equations for an IHS fluid, started in a uniform equilibrium state with temperature T 0 , admit an HCS solution (see, e.g., [2,3,7]) with a homogeneous temperature T ͑t͒, described by ≠ t T 22g 0 vT. Here the collision frequency is v͑T͒ ϳ p T͞l 0 with a mean free path l 0 ,given by the Enskog theory [8] for a dense system of hard disks or spheres (d 2, 3) and g 0 e͞2d. Then T ͑t͒ T 0 ͓͞1 1 g 0 v͑T 0 ͒t͔ 2 T 0 exp͑22g 0 t͒, where t is the average number of collisions suffered per particle within a time t. It is found by integrating dt v͑T ͑t͒͒dt. Moreover, this HCS solution is linearly unstable once the linear extent L of the system exceeds some dynamic correlation length, which increases with decreasing e, and is ...

Abstract. -The total energy E(t) in a fluid of inelastic particles is dissipated through inelastic collisions. When such systems are prepared in a homogeneous initial state and evolve undriven, E(t) decays initially as t −2 ∼ exp[−2ǫτ ] (known as Haff's law), where τ is the average number of collisions suffered by a particle within time t, and ǫ = 1 − α 2 measures the degree of inelasticity, with α the coefficient of normal restitution. This decay law is extended for large times to E(t) ∼ τ −d/2 in d-dimensions, far into the nonlinear clustering regime. The theoretical predictions are quantitatively confirmed by computer simulations, and holds for small to moderate inelasticities with 0.6 < α < 1.A fluid of inelastic hard spheres (IHS) in d dimensions, prepared initially in a state of thermal equilibrium with temperature T 0 , will remain for an extended period of time (measured in average number of collisions suffered per particle) in a spatially homogeneous cooling state (HCS), provided the degree of inelasticity ǫ is small. In this state the average kinetic energy per particle E(t) = d 2 T (t) is decreasing like 1/t 2 due to inelastic collisions, as first explained by Haff [1]. However, the HCS with a spatially uniform density and temperature, and a vanishing flow field is unstable against long wavelength spatial fluctuations, and leads finally to a state with large scale clusters in the density field n(r, t) as well as in the flow field u(r, t) (vortices). This is illustrated in the MD simulations of fig. 1. In this state cooling slows down and large deviations from Haff's law occur, which have not yet been explained in any quantitative manner [2, 3].In general analytic results for nonlinear decay of rapid granular flows are rare [2]. The goal of this letter is to calculate the long time decay of E(t) from nonlinear hydrodynamics, using mode coupling methods. The basic idea is to decompose the quantities of interest, here the total microscopic energy, or-in the study of long time tails of Green-Kubo formulas-the total microscopic momentum-, energy-or particle fluxes, into a superposition of products of slow hydrodynamic fluctuations of density n(r, t), temperature T (r, t) and flow field u(r, t).

We study the velocity distribution function for inelastic Maxwell models, characterized by a Boltzmann equation with constant collision rate, independent of the energy of the colliding particles. By means of a nonlinear analysis of the Boltzmann equation, we find that the velocity distribution function decays algebraically for large velocities, with exponents that are analytically calculated.

The solutions of the homogeneous nonlinear Boltzmann equation for inelastic Maxwell models, when driven by different types of thermostats, show, in general, overpopulated high energy tails of the form ϳexp(Ϫac), with power law tails and Gaussian tails as border line cases. The results are compared with those for inelastic hard spheres, and a comprehensive picture of the long time behavior in freely cooling and driven inelastic systems is presented.

Confined granular fluids, placed in a shallow box that is vibrated vertically, can achieve homogeneous stationary states thanks to energy injection mechanisms that take place throughout the system. These states can be stable even at high densities and inelasticities allowing for a detailed analysis of the hydrodynamic modes that govern the dynamics of granular fluids. Analyzing the decay of the time correlation functions it is shown that there is a crossover between a quasielastic regime in which energy evolves as a slow mode, to a inelastic regime, with energy slaved to the other conserved fields. The two regimes have well differentiated transport properties and, in the inelastic regime, the dynamics can be described by a reduced hydrodynamics with modified longitudinal viscosity and sound speed. The crossover between the two regimes takes place at a wavevector that is proportional to the inelasticity. A two dimensional granular model, with collisions that mimic the energy transfers that take place in a confined system is studied by means of microscopic simulations. The results show excellent agreement with the theoretical framework and allows the validation of hydrodynamic-like models.